Prepare a water bath for whisking the egg whites: You need two stainless steel pots, one slightly larger than the other. Pour about 1 inch of water into the larger pot. Place the smaller pot in the lower larger pot; the bottom of the upper pot must not touch the water surface. Place both pots on the stove and turn it on to high heat. As soon as the water starts to boil, the rising steam heats the upper pot. Once the pot is hot, turn the heat down to a minimum and let the water just simmer.
Pour the egg whites into the top pot and start whisking with a handheld electric mixer. Whisk slowly at first, then increase the speed. Add the coarse sugar at the same time. Keep an eye on the temperature in the pot, so it's not too high; the egg whites would start to boil. The temperature of the egg whites should be about so that when you put your finger in it, it will be pleasantly warm.
Whisk until the sugar has dissolved and the white meringue mass is set and beautifully glossy.
Remove the top pot with meringue and fold in the shredded coconut. Stir the mixture by hand.
Take the baking paper and prepare the baking sheet: Lightly grease the surface of the baking sheet, especially around the perimeter. Place the paper on top and press it down. The fat base will make the paper stick well. Then grease the paper over the surface, and dust it thinly with flour. The coconut cookies will not stick to the bottom and can be removed from the paper after baking without any problem.
Using two spoons, form the meringue mixture into little snowballs - mounds about 1 and ½ inches (4 cm) in diameter.
Bake the cookies in a preheated oven at 340 °F (170 °C) for 5 minutes. Then turn the temperature down to 266 °F (130 °C) and bake for another 15 minutes until the cookies are lightly golden on top.